A057708 -id:A057708 - cp's OEIS frontend

A004094 Powers of 2 written backwards.

1, 2, 4, 8, 61, 23, 46, 821, 652, 215, 4201, 8402, 6904, 2918, 48361, 86723, 63556, 270131, 441262, 882425, 6758401, 2517902, 4034914, 8068838, 61277761, 23445533, 46880176, 827712431, 654534862, 219078635, 4281473701, 8463847412, 6927694924, 2954399858, 48196897171

Offset: 0

N. J. A. Sloane

Freeman Dyson believes that A014963(a(n)) <> 5 is true but cannot be proved, see link. - Reinhard Zumkeller, Jan 05 2005

T. D. Noe, Table of n, a(n) for n = 0..1000
Edge Foundation, Annual Question 2005
Richard Lipton, More on testing Dyson's conjecture (2014)
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.

Cf. A014963, A102382, A102383, A102384, A102385.
The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).
Cf. A004086 (read n backwards).
For indices of primes see A057708.

a004094 = a004086 . a000079  -- Reinhard Zumkeller, Apr 02 2014

[Seqint(Reverse(Intseq(2^n))): n in [0..35]]; // Vincenzo Librandi, Jan 22 2020

a:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||(2^n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 21 2020

Table[FromDigits[Reverse[IntegerDigits[2^n]]], {n, 0, 35}] (* Vincenzo Librandi, Jan 22 2020 *)

rev(n)=subst(Polrev(digits(n)),'x,10)
a(n)=rev(2^n) \\ Charles R Greathouse IV, Oct 20 2014

apply( {A004094(n)=fromdigits(Vecrev(digits(2^n)))}, [0..44]) \\ M. F. Hasler, Feb 18 2021

def A004094(n):
    return int(str(2**n)[::-1]) # Chai Wah Wu, Feb 19 2021

a(n) = A004086(A000079(n)). - Reinhard Zumkeller, Apr 02 2014

More terms from Reinhard Zumkeller, Jan 05 2005

A058993 Numbers m such that 5^m reversed is a prime.

Original entry on oeis.org

1, 3, 26, 36, 43, 66, 76, 149, 511, 885, 3767, 18157, 20516, 26316

Offset: 1

Robert G. Wilson v, Jan 17 2001

Cf. A059206, A071583.

Numbers m such that k^m reversed is prime: A057708 (k=2), A350441 (k=4), this sequence (k=5), A058994 (k=7), A350442 (k=8), A058995 (k=13).

Do[ If[ PrimeQ[ ToExpression[ StringReverse[ ToString[5^n] ] ] ], Print[n] ], {n, 1, 3000} ]

isok(m) = isprime(fromdigits(Vecrev(digits(5^m)))) \\ Mohammed Yaseen, Jul 20 2022

from sympy import isprime
k, m, A058993_list = 1, 5,  []
while k <= 10**3:
    if isprime(int(str(m)[::-1])):
        A058993_list.append(k)
    k += 1
    m *= 5 # Chai Wah Wu, Mar 09 2021

a(12)-a(14) from Michael S. Branicky, Apr 01 2023

A102385 Primes in A004094.

Original entry on oeis.org

2, 23, 61, 821, 4201, 270131, 61277761, 274359834731, 23888027348153, 86936981079782063, 4243031147170261950811, 272838646828154727511151, 821882010875193363312928672952261, 274562423560500997742392394025368175422471, 4266472836315949449695828501889595072967427241

Offset: 1

Reinhard Zumkeller, Jan 06 2005

			n=17 -> 2^17 = 131072 -> 270131 = A000040(23652), therefore A004094(17) = 270131 is a term.

Jeozadaque Marcos da Silva, Table of n, a(n) for n = 1..22

Cf. A000040, A004094, A057708.

seq[digmax_] := Sort[Select[Table[IntegerReverse[2^n], {n, 0, Floor[digmax*Log2[10]]}], PrimeQ]]; seq[33] (* Amiram Eldar, Oct 02 2024 *)

A058994 Numbers m such that 7^m reversed is prime.

Original entry on oeis.org

1, 12, 24, 225, 392, 819, 1201, 1645, 1775, 37578

Offset: 1

Robert G. Wilson v, Jan 17 2001

Cf. A059208, A071584.

Numbers m such that k^m reversed is prime: A057708 (k=2), A350441 (k=4), A058993 (k=5), this sequence (k=7), A350442 (k=8), A058995 (k=13).

Do[ If[ PrimeQ[ ToExpression[ StringReverse[ ToString[7^n] ] ] ], Print[n] ], {n, 1, 2500} ]

isok(m) = isprime(fromdigits(Vecrev(digits(7^m)))) \\ Mohammed Yaseen, Jul 20 2022

from sympy import isprime
k, m, A058994_list = 1, 7,  []
while k <= 10**3:
    if isprime(int(str(m)[::-1])):
        A058994_list.append(k)
    k += 1
    m *= 7 # Chai Wah Wu, Mar 09 2021

a(10) from Michael S. Branicky, Mar 16 2025

A058995 Numbers m such that 13^m reversed is prime.

Original entry on oeis.org

1, 379, 467, 479, 1325, 7144, 10311, 26079

Offset: 1

Robert G. Wilson v, Jan 17 2001

Numbers m such that k^m reversed is prime: A057708 (k=2), A350441 (k=4), A058993 (k=5), A058994 (k=7), A350442 (k=8), this sequence (k=13).

Do[ If[ PrimeQ[ ToExpression[ StringReverse[ ToString[13^n] ] ] ], Print[n] ], {n, 1, 2300} ]
Select[Range[1400],PrimeQ[IntegerReverse[13^#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 18 2020 *)

isok(m) = isprime(fromdigits(Vecrev(digits(13^m)))) \\ Mohammed Yaseen, Jul 21 2022

from sympy import isprime
k, m, A058995_list = 1, 13,  []
while k <= 10**3:
    if isprime(int(str(m)[::-1])):
        A058995_list.append(k)
    k += 1
    m *= 13 # Chai Wah Wu, Mar 09 2021

a(6)-a(7) from Chai Wah Wu, Mar 09 2021

a(8) from Michael S. Branicky, Apr 01 2023

A085298 a(n) is the smallest exponent x such that prime(n)^x when reversed is a prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 8, 7, 1, 1, 2, 5, 15, 10, 12, 4, 39, 1, 1, 1, 11, 2, 1, 1, 10, 1, 23, 1, 5, 1, 243, 2, 1, 1, 1, 23, 1, 34, 1, 1, 1, 2, 58, 1, 3, 9, 166, 17, 68, 8, 8, 3, 7, 5, 5, 2, 2, 2, 61, 11, 97, 1, 1, 10, 2, 1, 1, 41, 1, 1, 66, 1, 5, 1, 1, 2, 2, 8, 40, 2, 8, 19, 2, 2, 723

Offset: 1

Labos Elemer, Jun 24 2003

It is conjectured that for every n such exponent exists.

			a(n)=1 means that rev(prime(n)) is prime i.e. prime(n) is in A007500;
a(n)=2 means that rev(prime(n)^2) is prime but rev(prime(n)) is not, like n=8:p=19 and 91 is not a prime but rev[19^2]=rev[361]=163 is a prime;
For n, the first k exponent providing rev(prime(n)^k) prime can be quite large, like at n=87: rev(p(87)^723)=rev(449^723) is the first [probably] prime has 1918 decimal digits: 948......573.

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A000040, A057708, A058993, A056994, A059695, A003459, A007500, A055387, A061461, A068652.

a:= proc(n) local k, p; p:= ithprime(n); for k while not isprime((s->
      parse(cat(seq(s[-i], i=1..length(s)))))(""||(p^k))) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 04 2019

a[n_] := Block[{k = 1}, While[! PrimeQ@ FromDigits@ Reverse@ IntegerDigits[ Prime[n]^k], k++]; k]; Array[a, 87] (* Giovanni Resta, Sep 04 2019 *)

a(n) = {my(x=1, p=prime(n)); while (!ispseudoprime(fromdigits(Vecrev(digits(p^x)))), x++); x;} \\ Michel Marcus, Sep 04 2019

a(n) = Min{x; reversed(prime(n)^x) is a prime}.

A350441 Numbers m such that 4^m reversed is prime.

Original entry on oeis.org

2, 5, 12, 35, 75, 182, 828, 1002, 1063, 2168, 6345, 6920, 10054, 14444, 51465

Offset: 1

Mohammed Yaseen, Dec 31 2021

From Bernard Schott, Jan 30 2022: (Start)
If m is a term, then u = 2*m is a term of A057708, because 4^m = 2^(2*m). In fact, terms of this sequence here are half the even terms of A057708.
If m is a term that is multiple of 3, then k = 2*m/3 is a term of A350442, because 4^m = 8^(2m/3). First examples: m = 12, 75, 828, 1002, 6345, 51465, ... and corresponding k = 8, 50, 552, 668, 4230, 34310, ... (End)

Cf. A058996, A071582.

Cf. Numbers m such that k^m reversed is prime: A057708 (k=2), this sequence (k=4), A058993 (k=5), A058994 (k=7), A350442 (k=8), A058995 (k=13).

Select[Range[2200], PrimeQ[IntegerReverse[4^#]] &] (* Amiram Eldar, Dec 31 2021 *)

isok(m) = isprime(fromdigits(Vecrev(digits(4^m))))

from sympy import isprime
m = 4
for n in range (1, 2000):
    if isprime(int(str(m)[::-1])):
        print(n)
    m *= 4

a(11)-a(15) from Amiram Eldar, Dec 31 2021

A350442 Numbers m such that 8^m reversed is prime.

Original entry on oeis.org

8, 15, 50, 552, 668, 1011, 1163, 1215, 2199, 4230, 7231, 34310

Offset: 1

Mohammed Yaseen, Dec 31 2021

From Bernard Schott, Jan 30 2022: (Start)
If k is a term, then u = 3*k is a term of A057708, because 8^k = 2^(3k).
If k is an even term, then t = 3*k/2 is a term of A350441, because 8^k = 4^(3k/2). First examples: k = 8, 50, 552, 668, 4230, 34310, ... and corresponding t = 12, 75, 828, 1002, 6345, 51465, ... (End)

Cf. A059003, A071586.

Cf. Numbers m such that k^m reversed is prime: A057708 (k=2), A350441 (k=4), A058993 (k=5), A058994 (k=7), A058995 (k=13).

Select[Range[2200], PrimeQ[IntegerReverse[8^#]] &] (* Amiram Eldar, Dec 31 2021 *)

isok(m) = isprime(fromdigits(Vecrev(digits(8^m))))

from sympy import isprime
m = 8
for n in range (1, 2000):
    if isprime(int(str(m)[::-1])):
        print(n)
    m *= 8

a(10)-a(12) from Amiram Eldar, Dec 31 2021

A085300 a(n) is the least prime x such that when reversed it is a power of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 71, 163, 18258901387, 90367894271, 13, 73, 1861, 344800741, 34351783286302805384336021, 940315563074788471, 1886172359328147919771, 14854831

Offset: 1

Labos Elemer, Jun 24 2003

A006567 (after rearranging terms) and A002385 are subsequences. - Chai Wah Wu, Jun 02 2016

			a(14)=344800741 means that 147008443=43^5=p(14)^5, where 5 is the smallest such exponent;
a(19) has 82 decimal digits and if reversed equals 39th power of p(19)=67.

Chai Wah Wu, Table of n, a(n) for n = 1..86

Cf. A085298, A057708, A058993, A056994, A059695, A003459, A007500, A055387, A061461, A068652.

from sympy import prime, isprime
def A085300(n):
    p = prime(n)
    q = p
    while True:
        m = int(str(q)[::-1])
        if isprime(m):
            return(m)
        q *= p # Chai Wah Wu, Jun 02 2016

A085299 a(n) is the smallest number x such that A085298[x]=n, or 0 if no such number exists.

Original entry on oeis.org

1, 8, 47, 18, 14, 89, 10, 9, 48, 16, 23, 17, 168, 268, 15, 661, 50, 380, 84, 116, 360, 245, 29, 144, 345, 227, 785, 261, 148, 235, 691, 658, 638, 40, 1023, 674, 1529, 210, 19, 81, 181, 428, 170, 1130, 2322, 406, 600, 373, 958, 217

Offset: 1

Labos Elemer, Jun 24 2003

			a(13) = 168 means that 13 is the smallest exponent such that reversed[p(168)^13] = reversed[997^13] = 776831144302925059735912605306533496169
is prime if read in this direction and 13th prime-power if read backwards.

Cf. A085298, A057708, A058993, A056994, A059695, A003459, A007500, A055387, A061461, A068652.

cp's OEIS Frontend

A004094 Powers of 2 written backwards.

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A058993 Numbers m such that 5^m reversed is a prime.

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A102385 Primes in A004094.

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A058994 Numbers m such that 7^m reversed is prime.

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A058995 Numbers m such that 13^m reversed is prime.

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A085298 a(n) is the smallest exponent x such that prime(n)^x when reversed is a prime.

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Maple

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A350441 Numbers m such that 4^m reversed is prime.

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